Keep in mind that these theoretical conclusions about importance measures are based upon assumptions. One of the assumptions is that the cost of improvement is constant, both across components and within a component across reliability of that component. This is a rather strong assumption.

For example, consider the constant cost of improvement assumption across components by reflecting on the relative cost of improving a turbine blade in a jet engine versus improving a rubber O-ring. It is likely to be much more expensive to get a slight improvement out of a turbine blade redesign as compared to upgrading the quality of an O-ring.

Also, consider the constant cost of improvement assumption within a component across reliability improvement. At the early stages of component development, reliability is lower but incremental improvements are generally much cheaper to introduce than in the later stages of component development. Therefore, importance measures should be used only as guides to selecting which components should be considered for improvement, not as a hard and fast rule that generates fixed decisions.

It is clear from the OR gate and AND gate examples that system structure and events are combined to determine the relative importance of each event. Also, it is clear that different importance measures (Birnbaum, Criticality and Fussell-Vesely) can assign different relative importance to each event.

Typically, an engineer chooses a given importance measure, such as Birnbaum, and then ranks or sorts the basic events in the fault tree on the basis of this important measure’s value. Those events with the largest importance measure values are then improved to reduce the chance of their occurrence. This improvement could be either a re-design of the basic component or sub-assembly or the introduction of redundancy at the component or sub-assembly level.

One reasonable question remains: Is the rank order of the chosen importance measure determined by the system structure and the events?

To answer this question, consider a system of six basic events, A1, A2, A3, B1, B2 and B3:

Assume that all of the basic events have exponential distributions with the Mean Time Between Failures (MTBF) given in Table C-1:

Table C-2 shows the Birnbaum importance measures for the top gate of this fault tree at time t = 1:

Ordering: A3> A1=A2>B2=B3>B1

Therefore, improve A3 first, then A1, A2, etc.

Table C-3 shows the Birnbaum importance measures for the top gate of this fault tree at time t = 100:

Ordering: A1= A2>A3=B2=B3>B1

Therefore, improve A1 and A2 first, then A3, etc.

Thus, the system structure and events do not by themselves determine the rank order of the chosen importance measure. Because event probabilities are used in determining the importance measures, and because event probabilities can change with time, the ranking of basic events by the importance measure will change over time for the same system and events. In this example, the rankings changed as follows:

Rank Ordering at time t = 1: A3> A1=A2>B2=B3>B1

Rank Ordering at time t = 100: A1= A2>A3=B2=B3>B1

Now, it has been shown that the rank ordering of a given importance measure can be affected by:

Also, this last example illustrates the fact that for a given set of basic events, a given system and a given time of evaluation, the rank orderings of different importance measures can be different.

For example, from the previous window for time t = 100, the importance measure values in Table C-4 were found:

These importance measure values yield the following rank orderings:

Birnbaum Rank Ordering: A1=A2>A3=B2=B3>B1

Criticality Rank Ordering: A1=A2>B1>A3=B2=B3

Fussell-Vesely Rank Ordering: A1=A2>A3=B1>B2=B3

All three importance measures indicate that A1 and A2 should be the first and second choices for improvement. It is not so clear that B2 and B3 should be the last choices for improvement because only two of the three measures place them as least important. Even so, and would probably be the last events to target for improvement.

The events A3 and B1 are the middle candidates for improvement. Because is third ranked twice and fourth ranked once, it is probably the third choice for improvement. However, B1 is much more difficult to classify. Although B1 is sixth place for the Birnbaum importance measure, it is third place for the Criticality importance measure and the fourth place for the Fussell-Vesely importance measure. Given that B1 ranks in front of B2 and B3 on two out of the three importance measures, B1 would probably be the fourth choice for improvement.